# Lesson 5

Negative Exponents with Powers of 10

### Lesson Narrative

Sometimes in mathematics, extending existing theories to areas outside of the original definition leads to new insights and new ways of thinking. Students practice this here by extending the rules they have developed for working with powers to a new situation with negative exponents. The challenge then becomes to make sense of what negative exponents might mean. This type of reasoning appears again in high school when students extend the rules of exponents to make sense of exponents that are not integers.

In analogy to positive powers of 10 that describe repeated multiplication by 10, this lesson presents negative powers of 10 as repeated multiplication by $$\frac{1}{10}$$, leading ultimately to the rule $$10^{\text-n}= \frac{1}{10^n}$$. Students use repeated reasoning to generalize about negative exponents (MP8). Students create viable arguments and critique the reasoning of others when comparing and contrasting, for example, $$\left(10^{\text-2}\right)^3$$ and $$\left(10^{2}\right)^{\text-3}$$ (MP3). With this understanding of negative exponents, all of the exponent rules created so far are seen to be valid for any integer exponents.

### Learning Goals

Teacher Facing

• Describe (orally and in writing) how exponent rules extend to expressions involving negative exponents.
• Describe patterns in repeated multiplication and division with 10 and $\frac{1}{10}$, and justify (orally and in writing) that $10^{\text-n}=\frac{1}{10^n}$.

### Student Facing

Let’s see what happens when exponents are negative.

### Required Preparation

Create a visual display for $$10^{\text-n} = \frac{1}{10^n}$$. For an example of how the rule works, consider showing $$10^{\text-3} = \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = \frac{1}{10^3}$$.

### Student Facing

• I can use the exponent rules with negative exponents.
• I know what it means if 10 is raised to a negative power.

Building On

### Glossary Entries

• base (of an exponent)

In expressions like $$5^3$$ and $$8^2$$, the 5 and the 8 are called bases. They tell you what factor to multiply repeatedly. For example, $$5^3$$ = $$5 \boldcdot 5 \boldcdot 5$$, and $$8^2 = 8 \boldcdot 8$$.

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